Black hole entropy as a Noether Charge

In summary, the conversation is about a paper by Thomas Mohaupt titled "Black hole entropy, special geometry and strings". The author defines entropy as a surface charge using a method by Wald, and considers a variation of the energy-momentum (E-M) tensor and the metric. The conversation also discusses how to write down the variation of the Lagrangian with respect to the Riemanntensor, and the definition of a current using Noethers theorem. There are questions about using test functions and partial derivatives in calculations, and recommendations for a good resource on the topic.
  • #1

haushofer

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Hi folks, I have a question about a paper by Thomas Mohaupt, called
"Black hole entropy, special geometry and strings". It's available here:

http://www.citebase.org/fulltext?format=application%2Fpdf&identifier=oai%3AarXiv.org%3Ahep-th%2F0007195 [Broken]

My question concerns part 2.2.5 , page 18. I find it quite difficult to get a nice feeling for doing calculations like the ones in that part. Here the author defines entropy as a surface charge ( a method due to Wald ). He assumes that the Lagrangian depends on the Riemanntensor, the energy-momentum (E-M) tensor and the derivative of the E-M tensor ( the covariant, I presume? ). He then considers a variation of the E-M tensor and the metric, which I do understand ( they're defined via a general coordinate transformation, so you end up with a Lie-derivative ) So up to that it's okay.

First, I want to do the variation of the Lagrangian with respect to the Riemanntensor, and here I get some troubles. How do I write down such a variation ? It looks like

[tex] \delta S = \frac{\partial L }{\partial R_{\mu\nu\rho\sigma} } \delta R_{\mu\nu\rho\sigma} [/tex]

Here L is the Lagrangian density.
If I want to rewrite this in terms of the variation with respect to the metric, can I use the usual chain rules for differentiation? Or should I do the differentiation explicitly with respect to the metric, and rewrite this in terms of the Riemanntensor?

Then they define a current J. Via Noethers theorem I know that one can define such a current as

[tex] J^{\mu} = \frac{\partial L}{\partial \phi ,\mu} \delta \phi [/tex]

where we sum over all fields phi. So my guess would be that the current which is written down there in section 2.2.5 is acquired via

[tex] J^{\mu} = \frac{\partial L}{\partial R_{\nu\lambda\rho\sigma},\mu} \delta R_{\nu\lambda\rho\sigma} + \frac{\partial L}{\partial \psi_{\nu\lambda},\mu} \delta \psi_{\nu\lambda} [/tex]

Is this going into the right direction? In the variation of the metric and the E-M tensor they use a test function epsilon which is a function of the coordinates, but I don't see it in the current back. That's strange, because you can't divide the function out of the current ( we get derivatices of the test function ). Or can we put those terms to 0? Also I have some doubts about when to use partial derivatives in such calculations, and when to use covariant derivatives. Should I just replace al partial derivatives in such calculations by covariant derivatives?

And does any-one know a good text(book) in which all this is nicely explained? I've been searching on the internet, but without good results. A lot of questions, I hope some-one can help me. Many thanks in forward,

Haushofer.
 
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Black hole entropy is a fundamental concept in the study of black holes and their thermodynamics. It is closely related to the Noether charge, which is a conserved quantity associated with a symmetry of a system. In the case of black holes, the Noether charge is related to the symmetries of the black hole's horizon.

The paper by Thomas Mohaupt, "Black hole entropy, special geometry and strings", discusses the calculation of black hole entropy using the Noether charge approach. In particular, part 2.2.5 focuses on the variation of the Lagrangian with respect to the Riemanntensor and the metric, and the definition of the current J.

To begin, the variation of the Lagrangian with respect to the Riemanntensor can be written as:

\delta S = \frac{\partial L}{\partial R_{\mu\nu\rho\sigma}} \delta R_{\mu\nu\rho\sigma}

where L is the Lagrangian density. To rewrite this in terms of the metric, you can use the chain rule for differentiation:

\frac{\partial L}{\partial R_{\mu\nu\rho\sigma}} = \frac{\partial L}{\partial g^{\alpha\beta}} \frac{\partial g^{\alpha\beta}}{\partial R_{\mu\nu\rho\sigma}}

Note that the metric is a function of the Riemanntensor, so you will need to use the inverse metric to find the second term in the above equation.

Next, the current J can be defined through Noether's theorem as:

J^{\mu} = \frac{\partial L}{\partial \phi ,\mu} \delta \phi

where \phi represents all the fields in the system. In the context of black hole entropy, the current is defined as:

J^{\mu} = \frac{\partial L}{\partial R_{\nu\lambda\rho\sigma},\mu} \delta R_{\nu\lambda\rho\sigma} + \frac{\partial L}{\partial \psi_{\nu\lambda},\mu} \delta \psi_{\nu\lambda}

where \psi_{\nu\lambda} represents the E-M tensor. This is the correct direction for the current, but note that the test function epsilon is not explicitly included. This is because the test function is taken to be
 

1. What is black hole entropy as a Noether charge?

Black hole entropy as a Noether charge is a concept in theoretical physics that relates to the information content of a black hole. It suggests that the entropy of a black hole is directly proportional to its surface area and is considered to be a conserved quantity, similar to energy or charge.

2. How is black hole entropy calculated as a Noether charge?

Black hole entropy as a Noether charge can be calculated using the black hole area theorem, which states that the surface area of a black hole can never decrease. This formula is based on the Noether charge, which is a conserved quantity that is associated with a symmetry of the system.

3. What is the significance of black hole entropy as a Noether charge?

The concept of black hole entropy as a Noether charge is significant because it provides a link between gravity and thermodynamics, two fundamental theories in physics. It also helps us understand the nature of black holes and their role in the universe.

4. Are there any unresolved issues with the concept of black hole entropy as a Noether charge?

While the concept of black hole entropy as a Noether charge has been widely accepted by the scientific community, there are still some unresolved issues and debates surrounding it. One of the main issues is the interpretation of the entropy as a Noether charge and its physical meaning.

5. How does black hole entropy as a Noether charge relate to the second law of thermodynamics?

The second law of thermodynamics states that the total entropy of a closed system always increases over time. Black hole entropy as a Noether charge provides a way to understand this law in the context of black holes, as it suggests that the entropy of a black hole is directly proportional to its surface area and can never decrease.

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